Given a monoidal category $(\mathcal{C}, \otimes, I)$ with coproducts, the free monoid on an object $A \in \mathcal{C}$ is usually constructed by first constructing the free pointed object on $A$, i.e. $I + A$, and then constructing the free monoid over the pointed object $I + A$. The free monoid construction here is understood as constructing the free monoid from a pointed object.
Why is it not constructed following the inverse factorisation? Why not first construct the free semigroup and then adjoint a point to a semigroup? Is this approach followed in some paper?